Model Selection

In this section, we will compare the performance of three different models: Linear, Exponential, and Log-Transformed for each fungal species at various temperatures. We will compare the following metrics:

Residual Sum of Squares (RSS): Lower values indicate a better fit.
Adjusted R-squared: Higher values indicate a better fit.
AIC (Akaike Information Criterion): Lower values indicate a better model fit considering model complexity.
BIC (Bayesian Information Criterion): Lower values indicate a better model fit, penalizing model complexity more heavily than AIC.

df <- read.csv("Rosalynn Tidy data.csv")
#Filter week 1-5, Suillus data up to week 6. 
df_1_5 <- df %>%
  filter(Week %in% 1:5)
# Convert relevant columns to factors
df_1_5$Temperature..Celsius. <- as.factor(df_1_5$Temperature..Celsius.)
#df_1_5$Week <- as.factor(df_1_5$Week)
df_1_5$Species <- as.factor(df_1_5$Species)
# Rename the column
colnames(df_1_5)[colnames(df_1_5) == "Temperature..Celsius."] <- "Temperature"

# Initialize an empty list to store model metrics
model_metrics <- list()

# Loop over each species and temperature to fit models and calculate metrics
for(species in unique(df_1_5$Species)) {
  for(temp in unique(df_1_5$Temperature)) {
    
    # Filter the data for this species and temperature
    df_subset <- df_1_5 %>% filter(Species == species, Temperature == temp)
    
    # Fit the Linear Model
    linear_model <- lm(Growth ~ Week, data = df_subset)
    
    # Fit the Exponential Model (log-transformed linear)
    exp_model <- lm(log(Growth) ~ Week, data = df_subset)
    
    # Fit the Logarithmic Model (Growth ~ log(Week))
    log_fit_model <- lm(Growth ~ log(Week), data = df_subset)
    
    # Calculate Metrics for each model
    metrics <- data.frame(
      Species = species,
      Temperature = temp,
      Metric = c("RSS", "Adjusted R-squared", "AIC", "BIC"),
      Linear_Model = c(sum(residuals(linear_model)^2),
                       summary(linear_model)$adj.r.squared,
                       AIC(linear_model),
                       BIC(linear_model)),
      Exponential_Model = c(sum(residuals(exp_model)^2),
                            summary(exp_model)$adj.r.squared,
                            AIC(exp_model),
                            BIC(exp_model)),
      Logarithmic_Model = c(sum(residuals(log_fit_model)^2),
                            summary(log_fit_model)$adj.r.squared,
                            AIC(log_fit_model),
                            BIC(log_fit_model))
    )
    
    # Determine the best model for each metric
    metrics$Best_Model <- sapply(1:nrow(metrics), function(i) {
      if (metrics$Metric[i] == "Adjusted R-squared") {
        # For Adjusted R-squared, higher is better
        if (metrics$Linear_Model[i] == max(metrics$Linear_Model[i], metrics$Exponential_Model[i], metrics$Logarithmic_Model[i])) {
          "Linear Model"
        } else if (metrics$Exponential_Model[i] == max(metrics$Linear_Model[i], metrics$Exponential_Model[i], metrics$Logarithmic_Model[i])) {
          "Exponential Model"
        } else {
          "Logarithmic Model"
        }
      } else {
        # For RSS, AIC, and BIC, lower is better
        if (metrics$Linear_Model[i] == min(metrics$Linear_Model[i], metrics$Exponential_Model[i], metrics$Logarithmic_Model[i])) {
          "Linear Model"
        } else if (metrics$Exponential_Model[i] == min(metrics$Linear_Model[i], metrics$Exponential_Model[i], metrics$Logarithmic_Model[i])) {
          "Exponential Model"
        } else {
          "Logarithmic Model"
        }
      }
    })
    
    # Store the metrics in the list
    model_metrics[[paste(species, temp, sep = "_")]] <- metrics
  }
}

# Combine all metrics into one data frame
model_comparison <- do.call(rbind, model_metrics)

# Display the table of model comparison with scrolling enabled for rows
model_comparison %>%
  kbl(caption = "Model Comparison for Different Species and Temperatures") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), 
                full_width = TRUE, 
                fixed_thead = TRUE) %>%
  scroll_box(height = "400px")

Model Comparison for Different Species and Temperatures
	Species	Temperature	Metric	Linear_Model	Exponential_Model	Logarithmic_Model	Best_Model
Suillus lakei_15.1	Suillus lakei	15	RSS	272.5171720	48.4593445	308.8850664	Exponential Model
Suillus lakei_15.2	Suillus lakei	15	Adjusted R-squared	0.4613154	0.4191081	0.3894270	Linear Model
Suillus lakei_15.3	Suillus lakei	15	AIC	378.5981283	212.8083901	390.6238204	Exponential Model
Suillus lakei_15.4	Suillus lakei	15	BIC	386.2911729	220.5014347	398.3168650	Exponential Model
Suillus lakei_20.1	Suillus lakei	20	RSS	1002.9973785	83.2938163	1119.0928300	Exponential Model
Suillus lakei_20.2	Suillus lakei	20	Adjusted R-squared	0.4608752	0.4044651	0.3984724	Linear Model
Suillus lakei_20.3	Suillus lakei	20	AIC	503.6905967	264.8067059	514.2050436	Exponential Model
Suillus lakei_20.4	Suillus lakei	20	BIC	511.3836413	272.4997505	521.8980882	Exponential Model
Suillus lakei_25.1	Suillus lakei	25	RSS	781.7938038	65.6273240	805.7073113	Exponential Model
Suillus lakei_25.2	Suillus lakei	25	Adjusted R-squared	0.4289472	0.4447257	0.4114798	Exponential Model
Suillus lakei_25.3	Suillus lakei	25	AIC	479.7715107	241.9220169	482.6639438	Exponential Model
Suillus lakei_25.4	Suillus lakei	25	BIC	487.4645552	249.6150614	490.3569883	Exponential Model
Rhizopogon Evadens_15.1	Rhizopogon Evadens	15	RSS	86.1196845	13.2576726	103.3863906	Exponential Model
Rhizopogon Evadens_15.2	Rhizopogon Evadens	15	Adjusted R-squared	0.5835773	0.6896881	0.5000860	Exponential Model
Rhizopogon Evadens_15.3	Rhizopogon Evadens	15	AIC	223.2203249	88.4966925	236.3772683	Exponential Model
Rhizopogon Evadens_15.4	Rhizopogon Evadens	15	BIC	230.0503232	95.3266909	243.2072666	Exponential Model
Rhizopogon Evadens_20.1	Rhizopogon Evadens	20	RSS	81.1488243	10.9114338	93.0074793	Exponential Model
Rhizopogon Evadens_20.2	Rhizopogon Evadens	20	Adjusted R-squared	0.7593021	0.7736558	0.7241278	Exponential Model
Rhizopogon Evadens_20.3	Rhizopogon Evadens	20	AIC	218.9396942	74.4735953	228.7601419	Exponential Model
Rhizopogon Evadens_20.4	Rhizopogon Evadens	20	BIC	225.7696926	81.3035936	235.5901403	Exponential Model
Rhizopogon Evadens_25.1	Rhizopogon Evadens	25	RSS	51.1903435	6.3616754	58.1441995	Exponential Model
Rhizopogon Evadens_25.2	Rhizopogon Evadens	25	Adjusted R-squared	0.8685156	0.8745631	0.8506544	Exponential Model
Rhizopogon Evadens_25.3	Rhizopogon Evadens	25	AIC	185.7668538	35.6281960	194.9378691	Exponential Model
Rhizopogon Evadens_25.4	Rhizopogon Evadens	25	BIC	192.5968522	42.4581944	201.7678674	Exponential Model
Rhizopogon idahoensis_15.1	Rhizopogon idahoensis	15	RSS	18.0173406	1.9029686	44.3805142	Exponential Model
Rhizopogon idahoensis_15.2	Rhizopogon idahoensis	15	Adjusted R-squared	0.9365388	0.9605924	0.8436818	Exponential Model
Rhizopogon idahoensis_15.3	Rhizopogon idahoensis	15	AIC	110.5832836	-51.2669267	175.4888245	Exponential Model
Rhizopogon idahoensis_15.4	Rhizopogon idahoensis	15	BIC	117.4132820	-44.4369284	182.3188228	Exponential Model
Rhizopogon idahoensis_20.1	Rhizopogon idahoensis	20	RSS	141.0385235	8.7132622	147.8315730	Exponential Model
Rhizopogon idahoensis_20.2	Rhizopogon idahoensis	20	Adjusted R-squared	0.8269137	0.8564575	0.8185771	Exponential Model
Rhizopogon idahoensis_20.3	Rhizopogon idahoensis	20	AIC	258.7375692	58.2761187	262.1244878	Exponential Model
Rhizopogon idahoensis_20.4	Rhizopogon idahoensis	20	BIC	265.5675675	65.1061171	268.9544862	Exponential Model
Rhizopogon idahoensis_25.1	Rhizopogon idahoensis	25	RSS	146.0640837	6.1040871	220.0835233	Exponential Model
Rhizopogon idahoensis_25.2	Rhizopogon idahoensis	25	Adjusted R-squared	0.9028412	0.9082441	0.8536050	Exponential Model
Rhizopogon idahoensis_25.3	Rhizopogon idahoensis	25	AIC	261.2584610	32.6522046	290.7757012	Exponential Model
Rhizopogon idahoensis_25.4	Rhizopogon idahoensis	25	BIC	268.0884593	39.4822030	297.6056996	Exponential Model
Cenococcum geophilum_15.1	Cenococcum geophilum	15	RSS	7.1878215	0.7879639	11.8191430	Exponential Model
Cenococcum geophilum_15.2	Cenococcum geophilum	15	Adjusted R-squared	0.7445867	0.9157614	0.5800165	Exponential Model
Cenococcum geophilum_15.3	Cenococcum geophilum	15	AIC	45.1730730	-7.8835153	57.1090498	Exponential Model
Cenococcum geophilum_15.4	Cenococcum geophilum	15	BIC	48.7072345	-4.3493538	60.6432113	Exponential Model
Cenococcum geophilum_20.1	Cenococcum geophilum	20	RSS	27.5942390	0.2729779	60.0069928	Exponential Model
Cenococcum geophilum_20.2	Cenococcum geophilum	20	Adjusted R-squared	0.8584880	0.9863473	0.6922651	Exponential Model
Cenococcum geophilum_20.3	Cenococcum geophilum	20	AIC	77.4583261	-33.3249922	96.1028241	Exponential Model
Cenococcum geophilum_20.4	Cenococcum geophilum	20	BIC	80.9924876	-29.7908307	99.6369856	Exponential Model
Cenococcum geophilum_25.1	Cenococcum geophilum	25	RSS	26.8718899	0.7088069	66.0314738	Exponential Model
Cenococcum geophilum_25.2	Cenococcum geophilum	25	Adjusted R-squared	0.9126857	0.9711796	0.7854453	Exponential Model
Cenococcum geophilum_25.3	Cenococcum geophilum	25	AIC	76.8216958	-10.4243741	98.3989138	Exponential Model
Cenococcum geophilum_25.4	Cenococcum geophilum	25	BIC	80.3558573	-6.8902126	101.9330753	Exponential Model
Laccaria bicolor_15.1	Laccaria bicolor	15	RSS	10.1641654	2.2452263	31.1866296	Exponential Model
Laccaria bicolor_15.2	Laccaria bicolor	15	Adjusted R-squared	0.9828993	0.9055001	0.9475299	Linear Model
Laccaria bicolor_15.3	Laccaria bicolor	15	AIC	53.4885979	17.2471093	80.3955048	Exponential Model
Laccaria bicolor_15.4	Laccaria bicolor	15	BIC	57.0227594	20.7812708	83.9296663	Exponential Model
Laccaria bicolor_20.1	Laccaria bicolor	20	RSS	32.5923101	2.3459816	20.3178084	Exponential Model
Laccaria bicolor_20.2	Laccaria bicolor	20	Adjusted R-squared	0.9604422	0.8754175	0.9753399	Logarithmic Model
Laccaria bicolor_20.3	Laccaria bicolor	20	AIC	81.4535907	18.3006513	70.1117040	Exponential Model
Laccaria bicolor_20.4	Laccaria bicolor	20	BIC	84.9877521	21.8348128	73.6458655	Exponential Model
Laccaria bicolor_25.1	Laccaria bicolor	25	RSS	12.7693784	1.0852276	43.0128546	Exponential Model
Laccaria bicolor_25.2	Laccaria bicolor	25	Adjusted R-squared	0.9772213	0.9364364	0.9232715	Linear Model
Laccaria bicolor_25.3	Laccaria bicolor	25	AIC	58.9649575	-0.2012898	88.1117340	Exponential Model
Laccaria bicolor_25.4	Laccaria bicolor	25	BIC	62.4991190	3.3328717	91.6458955	Exponential Model

Model Building: Exponential Models

In this section, we build exponential models to describe the growth of fungi at different temperatures. We will visualize both the raw data and the predicted data from the exponential models, and offer the option to compare them in the same plot.

# Create a list to store the fitted models and predicted data
exponential_models <- list()
predicted_data_list <- list()

# Loop over each species and temperature to fit exponential models
for(species in unique(df_1_5$Species)) {
  for(temp in unique(df_1_5$Temperature)) {
    
    # Filter the data for this species and temperature
    df_subset <- df_1_5 %>% filter(Species == species, Temperature == temp)
    
    # Fit the Exponential Model (log-transformed linear)
    exp_model <- lm(log(Growth) ~ Week, data = df_subset)
    
    # Store the model
    model_name <- paste("Exp_Model", species, temp, sep = "_")
    exponential_models[[model_name]] <- exp_model
    
    # Generate predicted values
    predicted_log_growth <- predict(exp_model, newdata = df_subset)
    predicted_growth <- exp(predicted_log_growth)  # Back-transform to the original scale
    
    # Store predicted data in a data frame
    df_predicted <- df_subset %>%
      mutate(Predicted_Growth = predicted_growth)
    
    # Store the predicted data for later use
    predicted_data_list[[model_name]] <- df_predicted
  }
}

# Combine all predicted data into one data frame for easier use
all_predicted_data <- bind_rows(predicted_data_list)

# Print a sample of the predicted data
head(all_predicted_data)

##   Temperature       Species strain_ID Week Plate Growth Predicted_Growth
## 1          15 Suillus lakei   Suilla     1     1  0.499         0.721077
## 2          15 Suillus lakei   Suilla     1     2  0.712         0.721077
## 3          15 Suillus lakei   Suilla     1     3  0.724         0.721077
## 4          15 Suillus lakei   Suilla     1     4  0.673         0.721077
## 5          15 Suillus lakei   Suilla     1     5  0.717         0.721077
## 6          15 Suillus lakei   Suilla     1     6  0.577         0.721077

Compare Raw and Fitted Data

Exponential Model Summary for Each Species at Each Temperature

The table below summarizes the results of fitting exponential models for each fungal species at different temperatures. For each combination of species and temperature, the following details are provided:

F-Value: This is the F-statistic for the model, which assesses the overall significance of the model.
P-Value: The P-value indicates the significance of the model. A lower P-value (< 0.05) suggests that the model is statistically significant.
Degrees of Freedom: This represents the number of data points minus the number of parameters estimated by the model.
Model Equation: The fitted equation for the model, represented as a linear relationship between the logarithm of growth and time (weeks).

# Initialize a data frame to store the summary information
model_summary <- data.frame(Species = character(), 
                            Temperature = numeric(), 
                            F_Value = numeric(), 
                            P_Value = numeric(), 
                            Degrees_of_Freedom = character(), 
                            Equation = character(),
                            stringsAsFactors = FALSE)

# Loop over each species and temperature to fit exponential models and extract metrics
for(species in unique(df_1_5$Species)) {
  for(temp in unique(df_1_5$Temperature)) {
    
    # Filter the data for this species and temperature
    df_subset <- df_1_5 %>% filter(Species == species, Temperature == temp)
    
    # Fit the exponential model (log-transformed linear)
    exp_model <- lm(log(Growth) ~ Week, data = df_subset)
    
    # Extract model summary details
    model_summary_info <- summary(exp_model)
    
    # Extract F-value, P-value, and degrees of freedom
    f_value <- model_summary_info$fstatistic[1]
    p_value <- pf(f_value, model_summary_info$fstatistic[2], model_summary_info$fstatistic[3], lower.tail = FALSE)
    df1 <- model_summary_info$fstatistic[2]  # Numerator degrees of freedom
    df2 <- model_summary_info$fstatistic[3]  # Denominator degrees of freedom
    
    # Extract the model equation
    intercept <- coef(exp_model)[1]
    slope <- coef(exp_model)[2]
    equation <- paste0("log(Growth) = ", round(intercept, 4), " + ", round(slope, 4), " * Week")
    
    # Store the results in the data frame
    model_summary <- rbind(model_summary, data.frame(Species = species, 
                                                     Temperature = temp, 
                                                     F_Value = f_value, 
                                                     P_Value = p_value, 
                                                     Degrees_of_Freedom = paste(df1, df2, sep = ", "), 
                                                     Equation = equation))
  }
}

# Create a scrollable table
model_summary %>%
  kbl(caption = "Exponential Model Summary for Each Species at Each Temperature") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = TRUE, fixed_thead = TRUE) %>%
  scroll_box(height = "400px")

Exponential Model Summary for Each Species at Each Temperature
	Species	Temperature	F_Value	Degrees_of_Freedom	Equation
value	<i>S. lakei</i>	15	69.54162	1, 94	log(Growth) = -0.7402 + 0.4132 * Week
value1	<i>S. lakei</i>	20	65.52046	1, 94	log(Growth) = -0.7776 + 0.5258 * Week
value2	<i>S. lakei</i>	25	77.08663	1, 94	log(Growth) = -0.6257 + 0.5062 * Week
value3	<i>R. evadens</i>	15	158.80206	1, 70	log(Growth) = -0.8093 + 0.437 * Week
value4	<i>R. evadens</i>	20	243.68149	1, 70	log(Growth) = -0.5387 + 0.4911 * Week
value5	<i>R. evadens</i>	25	496.02153	1, 70	log(Growth) = -0.5558 + 0.535 * Week
value6	<i>R. idahoensis</i>	15	1731.68367	1, 70	log(Growth) = -0.813 + 0.5467 * Week
value7	<i>R. idahoensis</i>	20	424.62689	1, 70	log(Growth) = -0.424 + 0.5793 * Week
value8	<i>R. idahoensis</i>	25	703.79203	1, 70	log(Growth) = -0.3219 + 0.6242 * Week
value9	<i>C. geophilum</i>	15	251.03408	1, 22	log(Growth) = -0.9864 + 0.4138 * Week
value10	<i>C. geophilum</i>	20	1662.65378	1, 22	log(Growth) = -1.0123 + 0.6269 * Week
value11	<i>C. geophilum</i>	25	776.04545	1, 22	log(Growth) = -0.9609 + 0.6901 * Week
value12	<i>L. bicolor</i>	15	221.38663	1, 22	log(Growth) = -0.2943 + 0.656 * Week
value13	<i>L. bicolor</i>	20	162.61667	1, 22	log(Growth) = 0.2903 + 0.5747 * Week
value14	<i>L. bicolor</i>	25	339.84252	1, 22	log(Growth) = 0.0793 + 0.5651 * Week

Interactive Exponential Growth Curves for Each Species at Each Temperature

The plot below displays the perfect exponential growth curves for each fungal species at different temperatures. These curves are based on the exponential model fitted to the data, using the equation:

\[ \text{Growth} = e^{(\text{intercept} + \text{slope} \times \text{Week})} \]

Calculation of the Derivative Function for the Exponential Model

For each fungal species at each temperature, an exponential growth model was fitted to the data. The general form of the model is:

\[ \text{Growth} = e^{(\text{intercept} + \text{slope} \times \text{Week})} \]

This model describes how the growth of the fungi changes over time. To determine how fast the growth changes with respect to time (week), we need to compute the derivative of the growth function with respect to the week.

Step 1: Apply the Chain Rule

To compute the derivative of the growth function, we apply the chain rule. The chain rule tells us how to differentiate composite functions like this one, where we have an exponent that depends on another variable (week).

We begin by differentiating the exponential function:

\[ \frac{d}{d\text{Week}} \left( e^{(\text{intercept} + \text{slope} \times \text{Week})} \right) \]

The chain rule gives us two parts: 1. The derivative of the exponential function, which is the same exponential function: \[ e^{(\text{intercept} + \text{slope} \times \text{Week})} \] 2. The derivative of the exponent \((\text{intercept} + \text{slope} \times \text{Week})\) with respect to week. The derivative of this linear term is just the slope: \[ \frac{d}{d\text{Week}} (\text{intercept} + \text{slope} \times \text{Week}) = \text{slope} \]

Step 2: Combine the Results

Multiplying the results of these two derivatives (following the chain rule), we get:

\[ \frac{d(\text{Growth})}{d(\text{Week})} = \text{slope} \times e^{(\text{intercept} + \text{slope} \times \text{Week})} \]

Step 3: Substitution

We know from the original model that:

\[ \text{Growth} = e^{(\text{intercept} + \text{slope} \times \text{Week})} \]

So, we can substitute this back into the derivative equation. Replacing the exponential term with Growth, we arrive at the simplified form:

\[ \frac{d(\text{Growth})}{d(\text{Week})} = \text{Growth} \times \text{slope} \]

What the Derivative Tells Us:

The derivative function tells us the rate of growth at any given point in time (week).
It shows that the rate of growth is proportional to the current growth value, scaled by the slope of the model.
As the fungi grow larger (as Growth increases), their growth rate also increases, but it is controlled by the value of the slope.

In the table below, for each species and temperature combination, the derivative is expressed in the simplified form:

\[ \frac{d(\text{Growth})}{d(\text{Week})} = \text{Growth} \times \text{slope} \]

This equation provides a clear understanding of how the rate of growth evolves over time for each species at each temperature.

# Initialize a data frame to store the simplified derivative functions
derivative_summary <- data.frame(Species = character(), 
                                 Temperature = numeric(), 
                                 Derivative_Function = character(),
                                 stringsAsFactors = FALSE)

# Loop over each species and temperature to compute simplified derivative functions
for(species in unique(df_1_5$Species)) {
  for(temp in unique(df_1_5$Temperature)) {
    
    # Filter the data for this species and temperature
    df_subset <- df_1_5 %>% filter(Species == species, Temperature == temp)
    
    # Fit the exponential model (log-transformed linear)
    exp_model <- lm(log(Growth) ~ Week, data = df_subset)
    
    # Extract the slope of the model
    slope <- coef(exp_model)[2]
    
    # The simplified derivative function is Growth * slope
    derivative_function <- paste0("d(Growth)/d(Week) = Growth * ", round(slope, 4))
    
    # Store the derivative function for this species and temperature
    derivative_summary <- rbind(derivative_summary, 
                                data.frame(Species = species, 
                                           Temperature = temp, 
                                           Derivative_Function = derivative_function))
  }
}

# Display the final table
library(knitr)
library(kableExtra)

# Create a scrollable table
derivative_summary %>%
  kbl(caption = "Simplified Derivative Functions of Exponential Models for Each Species at Each Temperature") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"), full_width = TRUE, fixed_thead = TRUE) %>%
  scroll_box(height = "400px")

Simplified Derivative Functions of Exponential Models for Each Species at Each Temperature
Species	Temperature	Derivative_Function
<i>S. lakei</i>	15	d(Growth)/d(Week) = Growth * 0.4132
<i>S. lakei</i>	20	d(Growth)/d(Week) = Growth * 0.5258
<i>S. lakei</i>	25	d(Growth)/d(Week) = Growth * 0.5062
<i>R. evadens</i>	15	d(Growth)/d(Week) = Growth * 0.437
<i>R. evadens</i>	20	d(Growth)/d(Week) = Growth * 0.4911
<i>R. evadens</i>	25	d(Growth)/d(Week) = Growth * 0.535
<i>R. idahoensis</i>	15	d(Growth)/d(Week) = Growth * 0.5467
<i>R. idahoensis</i>	20	d(Growth)/d(Week) = Growth * 0.5793
<i>R. idahoensis</i>	25	d(Growth)/d(Week) = Growth * 0.6242
<i>C. geophilum</i>	15	d(Growth)/d(Week) = Growth * 0.4138
<i>C. geophilum</i>	20	d(Growth)/d(Week) = Growth * 0.6269
<i>C. geophilum</i>	25	d(Growth)/d(Week) = Growth * 0.6901
<i>L. bicolor</i>	15	d(Growth)/d(Week) = Growth * 0.656
<i>L. bicolor</i>	20	d(Growth)/d(Week) = Growth * 0.5747
<i>L. bicolor</i>	25	d(Growth)/d(Week) = Growth * 0.5651

Rate of Growth Curves Based on the Derivative of the Exponential Model

Raw Growth Rate

The raw growth rate was calculated for each fungal strain on each plate, at every temperature and time point (week). This method provides the growth rate for each individual replicate, without aggregating across plates. The growth rate measures how much the fungal colony’s area increased between consecutive time points.

The growth rate was computed using the following formula:

\[ \text{Growth Rate} = \frac{\text{Growth at Week}_n - \text{Growth at Week}_{n-1}}{\text{Week}_n - \text{Week}_{n-1}} \]

###Aggregated Raw Growth Rate for Each Species at Different Temperatures

This figure displays the aggregated raw growth rates for multiple fungal species grown at various temperatures (15°C, 20°C, and 25°C). The growth rate is calculated based on the change in the mean growth between consecutive weeks, aggregated across multiple replicate plates for each species at each temperature.

Generated Niche Curve

This niche curve represents the growth rate at week 5 (The highest growth rate) for various fungal species grown at different temperatures (15°C, 20°C, and 25°C). The growth rate was calculated using the derivative of an exponential growth model fitted to each species’ growth data.

# Step 1: Fit an exponential model for each species and temperature, and calculate the growth rate at Week 5
df_growth_rate_week5 <- data.frame(Species = character(), 
                                   Temperature = numeric(), 
                                   Growth_Rate_Week5 = numeric(), 
                                   stringsAsFactors = FALSE)

for(species in unique(df_1_5$Species)) {
  for(temp in unique(df_1_5$Temperature)) {
    
    # Filter the data for this species and temperature
    df_subset <- df_1_5 %>% filter(Species == species, Temperature == temp)
    
    # Fit the exponential model (log-transformed linear)
    exp_model <- lm(log(Growth) ~ Week, data = df_subset)
    
    # Extract coefficients for the model (intercept and slope)
    intercept <- coef(exp_model)[1]
    slope <- coef(exp_model)[2]
    
    # Calculate the growth at Week 5
    growth_at_week5 <- exp(intercept + slope * 5)  # Growth at week 5
    
    # Calculate the derivative (growth rate) at Week 5
    growth_rate_at_week5 <- growth_at_week5 * slope
    
    # Store the results in the dataframe
    df_growth_rate_week5 <- rbind(df_growth_rate_week5, 
                                  data.frame(Species = species, 
                                             Temperature = as.numeric(temp),  # Ensure Temperature is numeric
                                             Growth_Rate_Week5 = growth_rate_at_week5))
  }
}

# Step 2: Create the plot of maximum growth rate at week 5 for each temperature
plot_growth_rate_temp <- ggplot(df_growth_rate_week5, aes(x = Temperature, y = Growth_Rate_Week5, color = Species)) +
  geom_point(size = 4) +  # Plot points for growth rate at week 5
  geom_line(aes(group = Species), size = 1.2) +  # Connect points by species
  scale_x_continuous(breaks = c(15, 20, 25), labels = c("15°C", "20°C", "25°C")) +  # Label temperature in °C
  labs(title = "Niche Curve of Different Fungi",
       x = "Temperature (°C)", y = "Growth Rate at Week 5 (cm^2 per Week)") +
  theme_minimal() +
  theme(
    strip.text.x = element_text(size = 10),  # Adjust species label size
    axis.text = element_text(size = 10),     # Adjust axis text size
    axis.title = element_text(size = 12),    # Adjust axis title size
    plot.title = element_text(size = 14, face = "bold"),  # Adjust title size
    legend.position = "right"                # Keep the legend for species
  )

# Step 3: Convert to interactive plot using plotly
interactive_plot_growth_rate_temp <- ggplotly(plot_growth_rate_temp)

# Step 4: Display the interactive plot
interactive_plot_growth_rate_temp

Raw Aggregated Niche Curve

# Step 1: Aggregate the data by calculating the mean growth for each species, temperature, and week across plates
df_aggregated <- df_1_5 %>%
  group_by(Species, Temperature, Week) %>%
  summarize(
    Mean_Growth = mean(Growth, na.rm = TRUE)  # Calculate mean growth across plates
  ) %>%
  arrange(Species, Temperature, Week)

# Step 2: Calculate the growth rate using the aggregated data
df_aggregated_growth_rate <- df_aggregated %>%
  mutate(
    Previous_Week = lag(Week),  # Previous week for each species and temperature
    Previous_Mean_Growth = lag(Mean_Growth),  # Previous mean growth value
    Growth_Rate = ifelse(is.na(Previous_Week) | is.na(Previous_Mean_Growth), 0, 
                         (Mean_Growth - Previous_Mean_Growth) / (Week - Previous_Week))  # Calculate mean growth rate
  ) %>%
  ungroup()

# Step 3: Find the maximum growth rate for each species and temperature
df_max_growth_rate <- df_aggregated_growth_rate %>%
  group_by(Species, Temperature) %>%
  summarize(Max_Growth_Rate = max(Growth_Rate, na.rm = TRUE)) %>%
  mutate(Temperature = as.numeric(Temperature)) %>%  # Convert Temperature to numeric
  ungroup()

# Step 4: Create the plot of maximum growth rate for each temperature
plot_max_growth_rate_temp <- ggplot(df_max_growth_rate, aes(x = Temperature, y = Max_Growth_Rate, color = Species)) +
  geom_point(size = 4) +  # Plot points for max growth rate
  geom_line(aes(group = Species), size = 1.2) +  # Connect points by species
  scale_x_continuous(breaks = c(15, 20, 25), labels = c("15°C", "20°C", "25°C")) +  # Label temperature in °C
  labs(title = "Raw Niche Curve",
       x = "Temperature (°C)", y = "Maximum Growth Rate (cm^2 per Week)") +
  theme_minimal() +
  theme(
    axis.text = element_text(size = 10),     # Adjust axis text size
    axis.title = element_text(size = 12),    # Adjust axis title size
    plot.title = element_text(size = 14, face = "bold"),  # Adjust title size
    legend.position = "right"                # Keep the legend for species
  )

# Step 5: Convert to interactive plot using plotly
interactive_plot_max_growth_rate_temp <- ggplotly(plot_max_growth_rate_temp)

# Step 6: Display the interactive plot
interactive_plot_max_growth_rate_temp

Fungal Growth Analysis

Your Name

Model Selection

Model Building: Exponential Models

Compare Raw and Fitted Data

Exponential Model Summary for Each Species at Each Temperature

Interactive Exponential Growth Curves for Each Species at Each Temperature

Calculation of the Derivative Function for the Exponential Model

Step 1: Apply the Chain Rule

Step 2: Combine the Results

Step 3: Substitution

What the Derivative Tells Us:

Rate of Growth Curves Based on the Derivative of the Exponential Model

Raw Growth Rate

Generated Niche Curve

Raw Aggregated Niche Curve